AMC 10B/12B 2025 (Problem 24/20)

A frog hops along the number line according to the following rules.

It starts at $0$ .
If it is at $0$ , then it moves to $1$ with probability $\frac12$ and it disappears with probability $\frac12$ .
For $n=1,2,$ or $3$ , if it is at $n$ , then it moves to $n+1$ with probability $\frac14$ , it moves to $n-1$ with probability $\frac14$ , and it disappears with probability $\frac12$ .

What is the probability that the frog reaches

4

\text{(A)}\;\frac{1}{101}

\text{(B)}\;\frac{1}{100}

\text{(C)}\;\frac{1}{99}

\text{(D)}\;\frac{1}{98}

\text{(E)}\;\frac{1}{97}